Algebraic background for numerical methods, control theory and renormalization

TL;DR

This paper reviews various algebraic structures like Hopf algebras, operads, and pre-Lie algebras, highlighting their roles across diverse fields such as control theory, numerical analysis, and quantum field renormalization.

Contribution

It provides a comprehensive overview of algebraic frameworks connecting mathematics with applied areas like control, numerical methods, and quantum physics.

Findings

Connected Hopf algebras are fundamental in multiple fields.

Pre-Lie algebras play a significant role in the discussed areas.

Various related algebraic structures are interconnected and relevant.

Abstract

We review some important algebraic structures which appear in a priori remote areas of Mathematics, such as control theory, numerical methods for solving differential equations, and renormalization in Quantum Field Theory. Starting with connected Hopf algebras we will also introduce augmented operads, and devote a substantial part to pre-Lie algebras. Other related algebraic structures (Rota-Baxter and dendriform algebras, NAP algebras) will be also mentioned.